We introduce a new property $(gaR)$ extending the property $(R)$ considered by Aiena. We study the property $(gaR)$ in connection with Weyl type theorems and establish sufficient and necessary conditions under which the property $(gaR)$ holds. In addition, we also study the stability of the property $(gaR)$ under perturbations by finite-dimensional operators,
by nilpotent operators, by quasinilpotent operators, and by algebraic operators commuting with $T$. The classes of operators
are considered as illustrating examples.
